Schatten p-classes and asymptotics for eigenvalues of positive integral operators on the sphere

Abstract

Abstract: We obtain decay rates for eigenvalues and singular values of positive integral operators generated by square integrable kernels on the unit sphere in Rm+1, m ≥ 2, under assumptions on both, certain derivatives of the kernel and the integral operators generated by such derivatives. This type of problem is common in the literature but the assumptions are usually defined using standard differentiation in Rm+1. The rates we present depend on both, the differentiability order used to define the smoothness conditions and the dimension m. Some of them are shown to be optimal. The results we show here are published in [2] except Theorem 2.5 which extends the
main result from there.